Solve 2(x-5)^2=4(x-7)

$$2 ( x - 5 ) ^ { 2 } = 4 ( x - 7 )$$

Solve for x (complex solution)

$$x=6+\sqrt{3}i\approx 6+1.732050808i$$

$$x=-\sqrt{3}i+6\approx 6-1.732050808i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$2\left(x^{2}-10x+25\right)=4\left(x-7\right)$$

Use the distributive property to multiply 2 by x^{2}-10x+25.

$$2x^{2}-20x+50=4\left(x-7\right)$$

Use the distributive property to multiply 4 by x-7.

$$2x^{2}-20x+50=4x-28$$

Subtract 4x from both sides.

$$2x^{2}-20x+50-4x=-28$$

Combine -20x and -4x to get -24x.

$$2x^{2}-24x+50=-28$$

Add 28 to both sides.

$$2x^{2}-24x+50+28=0$$

Add 50 and 28 to get 78.

$$2x^{2}-24x+78=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and 78 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\times 78}}{2\times 2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\times 78}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\times 78}}{2\times 2}$$

Multiply -8 times 78.

$$x=\frac{-\left(-24\right)±\sqrt{576-624}}{2\times 2}$$

Add 576 to -624.

$$x=\frac{-\left(-24\right)±\sqrt{-48}}{2\times 2}$$

Take the square root of -48.

$$x=\frac{-\left(-24\right)±4\sqrt{3}i}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{3}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±4\sqrt{3}i}{4}$$

Now solve the equation x=\frac{24±4\sqrt{3}i}{4} when ± is plus. Add 24 to 4i\sqrt{3}.

$$x=\frac{24+4\sqrt{3}i}{4}$$

Divide 24+4i\sqrt{3} by 4.

$$x=6+\sqrt{3}i$$

Now solve the equation x=\frac{24±4\sqrt{3}i}{4} when ± is minus. Subtract 4i\sqrt{3} from 24.

$$x=\frac{-4\sqrt{3}i+24}{4}$$

Divide 24-4i\sqrt{3} by 4.

$$x=-\sqrt{3}i+6$$

The equation is now solved.

$$x=6+\sqrt{3}i x=-\sqrt{3}i+6$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$2\left(x^{2}-10x+25\right)=4\left(x-7\right)$$

Use the distributive property to multiply 2 by x^{2}-10x+25.

$$2x^{2}-20x+50=4\left(x-7\right)$$

Use the distributive property to multiply 4 by x-7.

$$2x^{2}-20x+50=4x-28$$

Subtract 4x from both sides.

$$2x^{2}-20x+50-4x=-28$$

Combine -20x and -4x to get -24x.

$$2x^{2}-24x+50=-28$$

Subtract 50 from both sides.

$$2x^{2}-24x=-28-50$$

Subtract 50 from -28 to get -78.

$$2x^{2}-24x=-78$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=-\frac{78}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=-\frac{78}{2}$$

Divide -24 by 2.

$$x^{2}-12x=-\frac{78}{2}$$

Divide -78 by 2.

$$x^{2}-12x=-39$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-39+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-39+36$$

Add -39 to 36.

$$x^{2}-12x+36=-3$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=-3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{-3}$$

Simplify.

$$x-6=\sqrt{3}i x-6=-\sqrt{3}i$$

Add 6 to both sides of the equation.

$$x=6+\sqrt{3}i x=-\sqrt{3}i+6$$